Jacobi’s Elliptic Functions

  • Derek F. Lawden
Part of the Applied Mathematical Sciences book series (AMS, volume 80)

Abstract

The elliptic functions sn u, cn u, and dn u are defined as ratios of theta functions as below:
$$sn\;u = \frac{{{\theta _3}(0)}}{{{\theta _2}(0)}}\cdot \frac{{{\theta _1}(z)}}{{{\theta _4}(z)}},$$
(2.1.1)
$$cn\,u = \frac{{{\theta _4}(0)}}{{{\theta _2}(0)}}\cdot \frac{{{\theta _2}(z)}}{{{\theta _4}(z)}},$$
(2.1.2)
$$dn\;u = \frac{{{\theta _4}(0)}}{{{\theta _3}(0)}}\cdot \frac{{{\theta _3}(z)}}{{{\theta _4}(z)}},$$
(2.1.3)
where z = u 3 2 (0). sn u is read as “es en yew” or as “san yew”; cn u and dn u can similarly be read letter by letter or as “can u” and “dan u” respectively.

Keywords

Elliptic Function Theta Function Multivalued Function Simple Pole Fundamental Period 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1989

Authors and Affiliations

  • Derek F. Lawden
    • 1
  1. 1.University of Aston in BirminghamBirminghamUK

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